3. Mean Field Theory¶
3.1. Essence of Mean Field Approximation (MFA): replacing fluctuating terms by averages¶
Let us assume that each spin i independently of each other feels some average effect of a field:
Each spin is experiencing a local field defined by its nearest neighours.
The difficulty with effective field is that \(H_i\) depends on the instatnaous states of neighbouring spins of \(s_i\) which flucutuate
We now make a dramaic approximations: replace the effective field by its mean field approximation where each spin is experiencing a field independent of others. The average magnetiszation per spin due to trasnlational invariance is same for every spin (in perioid cboundary conditions that is)
z=4 for 2D lattice
z=6 for 3D lattice
In MFA hamitlonian factors out into additive components
Just like the exact case we had with J=0.
The \(h=0\) MFA case
The equation can be solved in a self-consistent manner or graphically by finding intersection between:
\(m =tanh(x)\)
\(x = \beta Jzm\)
When the slope is equal to one it provides a dividing line between two behaviours.
MFA shows phase transitio for 1D Ising model at finite \(T=T_c\)!
import holoviews as hv
import ipywidgets as widgets
import matplotlib.pyplot as plt
import numpy as np
import scipy as sp
@widgets.interact(T=(0.1,5), Tc=(0.1,5))
def mfa_ising1d_plot(T=1, Tc=1):
x = np.linspace(-3,3,1000)
f = lambda x: (T/Tc)*x
m = lambda x: np.tanh(x)
plt.plot(x,m(x), lw=3, alpha=0.9, color='green')
plt.plot(x,f(x),'--',color='black')
idx = np.argwhere(np.diff(np.sign(m(x) - f(x))))
plt.plot(x[idx], f(x)[idx], 'ro')
plt.legend(['m=tanh(x)', 'x'])
plt.ylim(-2,2)
plt.grid('True')
plt.xlabel('m',fontsize=16)
plt.ylabel(r'$tanh (\frac{Tc}{T} m )$',fontsize=16)
@widgets.interact(Tc_T=(0.1,5))
def mfa_ising1d_plot(Tc_T=1):
x = np.linspace(-1,1,200)
h = lambda x: np.arctanh(x) - Tc_T*x
plt.plot(h(x),x, lw=3, alpha=0.9, color='green')
plt.plot(x, np.zeros_like(x), lw=1, color='black')
plt.plot(np.zeros_like(x), x, lw=1, color='black')
plt.grid(True)
plt.ylabel('m',fontsize=16)
plt.xlabel('h',fontsize=16)
plt.ylim([-1,1])
plt.xlim([-1,1])
3.1.1. Critical exponents¶
A signature of phase transitions and critical phenomena is that there are universal power law behaviours near critical point
Correlation lengths \(\xi\) diverge at critical points
3.1.2. Mean field exponents¶
We can derive the value of critical exponent \(\beta\) within mean field approximation by Taylor expanding the hyperbolic tangent
One solution is obviously m = 0 which is the only solution above \(T_c\)
Below \(T_c\) the non-zero solution is found \(m=\sqrt{3}\frac{T}{T_c} \Big(1-\frac{T}{T_c} \Big)^{1/2}+...\)
\(\beta_{MFA}=1/2\)
3.1.3. Helmholtz Free energy¶
We will now make use of Mean field theory to approximate dependence of field on magnetization: \(h(m) \approx m(1-T_c/T)+ 1/3 m^3\) which enables us to evaluate the integral above.
Equilibirum is found by minimizing the free energy: \(aM +bM^3 = 0\) with solutions M = 0 and \(M=\pm (-a/b)^{1/2}\)
\(T < T_c\) case we get \(a<0\) and \(M=\pm (-|a|/b)^{1/2} = \pm M_S\)
\(T > T_c\) case we get \(a>0\) and \(M=0\)
@widgets.interact(T=(400,800))
def HelmF(T=400):
Tc=631 # constnt for Ni
a = 882*(T/Tc-1)
b = 0.4734*T
M = np.linspace(-2,2,1000)
plt.plot(M, 0.5*a*M**2 + 0.25*b*M**4, lw=4, color='brown', label=f"T/Tc = {(T/Tc)}")
plt.grid(True)
plt.xlim([-2,2])
plt.ylim([-140,200])
plt.ylabel('$F(M)$')
plt.xlabel('$M$')
plt.legend()
hv.extension('plotly')
def F_m(T):
Tc = 631
a = 882*(T/Tc-1)
b = 0.4734*T
M = np.linspace(-2,2,1000)
free_M = 0.5*a*M**2 + 0.25*b*M**4
return hv.Curve((M, free_M)).opts(xlim=(-2, 2), ylim=(-140, 200))
hv.HoloMap({ T: F_m(T) for T in np.linspace(400, 800, 100)}, kdims=['T'])
3.1.4. Problems¶
Use Transfer matrix method to solve general 1D Ising model with \(h = 0\) (Do not simply copy the solution by setting h=0 but repeat the derivation :)
Find the free energy per particle \(F/N\) in the limit \(n\rightarrow \infty \) for both periodic bounary codnition conditions and free boundary cases.
Plot temperature dependence of heat capacity and free energy as a function for \((h=\neq, J\neq 0)\) \((h=0, J\neq 0)\) and \((h=\neq, J\neq \neq)\) cases of 1D Ising model. Coment on the observed behaviours.
Explain why heat capacity and magnetic susceptibility diverge at critical temperatures.
Explain why correlation functions diverge at a critical temperature
Explain why are there universal critical exponents.
Explain why the dimensionality and range of intereactions matters for existance and nature of phase transitions.
Using mean field approximation show that near critical temperature magnetization per spin goes as \(m\sim (T_c-T)^{\beta}\) (critical exponent not to nbe confused with inverse \(k_B T\)) and find the value of \beta. Do the same for magnetic susceptibility \(\chi \sim (T-T_c)^{-\gamma}\) and find the value of \(\gamma\)
Explain what is the nature of mean field theory approximation and why is predictions fail for low dimensionsal systems but consistently get better with higher dimensions?
Consider a 1D model given by the Hamiltonian:
where \(J>1\), \(D>1\) and \(s_i =-1,0,+1\)
Assuming periodic boundary codnitions calcualte eigenvalues of the transfer matrix
Obtain expresions for internal energy, entropy and free energy
What is the ground state of this model (T=0) as a function of \(d=D/J\) Obtain asymptotic form of the eigenvalues of the transfer matrix in the limit \(T\rightarrow 0\) in the characteristic regimes of d (e.g consider differnet extereme cases)